1 Answer
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Screen sharing and AirPlay can work over Bluetooth -- WiFi is not necessary. On the other hand the Remote app requires that the AppleTV is connected to WiFi.

This explained my problem. My AppleTV was connected to a 2.4 GHz WiFi network, which stopped working. For some reason the 5 GHz network continued to work and my iPhone switched over, so I didn't notice.

Since I didn't have an original remote, I was able to follow these steps to switch the AppleTV over to the 5 GHz network: